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IMPULSE AND MOMENTUM                      261
1.   A gun is free to move on smooth horizontal tracks.   Show that the
loss of energy due to recoil is        — E, where M and m are the masses of
the gun and the projectile respectively, and E is the kinetic energy which is transmitted to the gun and projectile.
2.   In the preceding problem compare the velocities of the projectile when the gun is fixed and when free to move.   Also show that the actual
angle a' at which the projectile leaves the gun is given by tan CL =      J" —
tan a, where a is the angle which the gun makes with the horizon.
3.   A man stands on a plank of mass m, which is on a perfectly smooth horizontal plane.    He jumps upon another plank of the same mass, then back upon the first plank.   Find the ratio of the velocities of the two planks if the mass of the man is M.
4.   A stream of water delivering 1000 gallons per minute, at a velocity
of 20 —-, strikes a plane (1) normally, (2) at an angle of 30°.   Find the sec.
force exerted on the plane.
6. A uniform chain is held coiled up close to the edge of a smooth table, with one end hanging over the edge. Discuss the motion of the chain when it is allowed to fall, supposing the part hanging over the edge to be very small at the start of the motion.
6.   In the preceding problem show that the acceleration is constant if the density of the chain varies as the distance from that end of the chain which is in motion.
7.   A mass of snow begins to slide down a regular slope, accumulating more snow as it moves along, thus forming an avalanche.    Supposing the path cleared to be of uniform depth and width, show that the acceleration of the avalanche is constant.
8.   A ball falls on a floor from a height h and rebounds each tune vertically.                                                       __
(a)  Show that                  T = f^ V/— >
L       6         Q
where T is the total time taken by the ball to come to rest.   Find the value of T for h = 25 feet and e = 0.5.
(b)  Show that              H = \ąŁh,
1 — &
where H is the total distance described.   Find the value of H for h = 25 feet and e = 0.5.